Relation between transport functor of a fibration and a Hurewicz connection on it This is a crosspost of this MSE question.

Let $A\overset{\alpha}{\rightarrow}B$ be a (Hurewicz) fibration.


*

*The homotopy lifting property w.r.t a fiber $\alpha ^{-1}(b)$
furnishes for each path $b\to b^\prime$ in the base a continuous map
$\alpha ^{-1}(b)\to \alpha ^{-1}(b^\prime)$. Moreover, this
assignment extends to a functor $\pi_1B\longrightarrow
   \mathsf{hTop}$.

*On the other hand, as a fibration $\begin{smallmatrix}A\\\downarrow\\B\end{smallmatrix}$ admits a Hurewicz connection $s$. Given such a connection it is tempting to send a path $b\overset{\gamma}{\to} b^\prime $ in the base to the following set function (analogously to covering space theory) $$\alpha^{-1}(b)\longrightarrow \alpha ^{-1}(b^\prime),\quad a\mapsto \operatorname{eval}_1s(a,\gamma).$$ I suspect this set function might be continuous, but I see no reason for it to be a homotopy equivalence, since $s(a,\gamma)$ need not be related in a nice way to lifts of opposite path $b\overset{\bar\gamma}{\leftarrow} b^\prime $.


Questions.


*

*Is fiber transport along a Hurewicz connection continuous?

*(Assuming continuity) Is fiber transport along a Hurewicz connection functorial?

*(Assuming continuity) Does it coincide with the first transfer functor?

*Suppose the fibers are all homeomorphic. Are there any interesting conditions that make the transport functor $\pi_1B\longrightarrow \mathsf{hTop}$ lift to $\mathsf{Top}$? That is, can we obtain such homeomorphisms via transport?



I thought about possible functoriality of the transport along the Hurewicz connection. We want $\operatorname{eval}_1(a,\delta \ast \gamma)=\operatorname{eval}_1s(\operatorname{eval}_1s(a,\gamma),\delta,)$. We may consider the concatenation $s(\operatorname{eval}_1s(a,\gamma),\delta)\ast s(a,\gamma)$ which seems to lift $\delta \ast \gamma$, but I'm not quite sure where to go from here.
 A: Let $p: E\to B$ be a map. Define $\Lambda(p) = E \times_B B^I$; this is the space of pairs $(x,\gamma)$ consisting of a point $x\in E$ and a path $\gamma: [0,1] \to B$ such that $\gamma(0) = p(x)$. There is an evident restriction map 
$$
\rho: E^I \to \Lambda(p)
$$
where $E^I$ is the free path space of $E$. 
The map $p: E\to B$ is a Hurewicz fibration iff $\rho$ has a section (easy exercise).  A choice of (continuous) section $s: \Lambda(p) \to E^I$ defines transport.  
Assume $p$ is a Hurewicz fibration.
Consider the commutative diagram
$\require{AMScd}$
$$
\begin{CD}
\Lambda(p) \times 0 @>>> E \\
@VVV @VV pV \\
\Lambda(p) \times I @>>> B
\end{CD}
$$
where the bottom arrow is given by $((x,\gamma),t) \mapsto \gamma(t)$ and
the top one is given by  $((x,\gamma),0)\mapsto x$. By the lifting property, we can fill in the diagram with a  map 
$$
\Lambda(p) \times I \to E \, .
$$
This continuous map defines the Hurewicz connection by taking its adjoint to get a map $\Lambda(p) \to E^I$.
On the other hand, for a given path $\gamma$ wiht $\gamma(0) = b$ and $\gamma(1) = b'$, we can restrict the displayed map $\Lambda(p) \times I \to E$ to 
$F_b = (F_b \times_{\{b\}} \{\gamma\}) \times \{1\}$ (with $F_b = p^{-1}(b)$)
to obtain a continuous map
$$
F_b \to E
$$ 
whose image is contained in $F_{b'}$. This map is your transport operation. This establishes (1)-(3).
As to (4), the transport functor restricts on closed based loops to a holonomy map
$$
\Omega B \to G(F)
$$
where $F$ is the fiber at the basepoint and $G(F)$ is the (group-like) topological monoid of self homotopy equivalences of $F$. This map is actually an $A_\infty$-homomorphism. By replacing $\Omega B$ with its Moore loops instead, this will become a morphism of topological monoids.
Let us make the additional hypothesis that your Hurewicz fibration is a fiber bundle (this is actually not a restriction up to fiber equivalence). Then your transport map gives a holonomy homomorphism
$$
\Omega B \to H(F) 
$$
where $H(F)$ is the topological group of self homeomorphisms of $F$.
Your question (4) is more-or-less the condition that the connection admit a flat reduction, meaning this: let  $H^\delta(F)$ be $H(F)$ with the discrete topology. Then we have a map of classifying spaces
$$
BH^\delta(F) \to BH(F) 
$$ and (4) is answered in the affirmative iff the classifying map for the bundle
$B\to BH(F)$ factors up to homotopy through $BH^\delta(F)$.
